Cytokine-Induced Injury of Articular
Cartilage: A Computational Study
1 Department of Mathematics, University of Scranton, Scranton, PA, USA
Abstract
In previous works, the author and collaborators establish a mathe-
matical model for injury response in articular cartilage. In this paper
we use mathematical software and computational techniques, applied
to an existing model to explore in more detail how the behavior of
cartilage cells is influenced by several of, what are believed to be, the
most significant mechanisms underlying cartilage injury response at
the cellular level. We introduce a control parameter, the radius of
attenuation, and present some new simulations that shed light on how
inflammation associated with cartilage injuries impacts the metabolic
activity of cartilage cells. The details presented in the work can help
to elucidate targets for more effective therapies in the preventative
treatment of post-traumatic osteoarthritis.
1 Background
Injury response and wound healing is a topic of central importance in biomed-
ical research for obvious reasons. As a result, there has been a great deal
of activity in developing mathematical and computational models of wound
healing in various organ systems such as skin, see e.g. [12]. In contrast, there
has been very little activity in developing computational models for wound
healing and injury response in articular cartilage, despite a great interest in
this topic in orthopaedics research. What is more, few if any of the mathe-
matical models developed for wound healing in other systems are appropriate
for application to articular cartilage.
Articular cartilage is made up of differentiated mesenchymal cells known
as chondrocytes. These cells are embedded in an extracellular matrix and
are responsible for the biomechanical properties of cartilage [16]. Mechanical
stress and injury influence changes in the metabolic activity of chondrocytes
[16]. Specifically, during injury response chondrocytes produce and respond
to certain cytokines, or signaling molecules, known as tumor necrosis factor
α (TNF-α) and erythropoietin (EPO). There is a “balancing act” between
2
the pro-inflammatory cytokine TNF-α and the anti-inflammatory cytokine
EPO in which each limits the production and biological action of the other.
In a recent article Brines and Cerami [3] suggest that TNF-α plays a sig-
nificant role in causing the spread of cartilage lesions, while EPO plays an
antagonistic role to TNF-α, limiting the area over which a lesion can spread
by counteracting some of the effects of inflammation [3]. It has also been
observed that there are inherent time-delays in the activation of, and signal-
ing by EPO that results in a window of opportunity for the spread of lesions
due to secondary injury caused by inflammation. However, the authors of
[3] suggest that it may be possible to intervene with EPO derived therapies
to minimize the amount of secondary injury due to inflammation and the
spread of cartilage damage.
In previous works [5, 6], the author, with collaborators, develop a novel
mathematical model for articular cartilage injury response aiming to test hy-
potheses put forth in [3]. As with any mathematical or computational model,
it is important to understand how the behavior of the results depend on the
parameter values. In particular, it is useful to know how changes in the pa-
rameter values effect the results of simulations. This is especially the case
when the goal is to tie the modeling efforts to experimental results. On the
one hand, any measurement involves an error and if the model is sensitive to
small changes in the parameter values, often the case when models contain
nonlinear terms, the experimental error may be significant enough that simu-
lations behave differently than what is to be expected, based on experimental
3
observations. On the other hand, different parameter values may correspond
to different types of observed behavior, or even more interestingly, changes
in parameter values can lead to predictions about the system being modeled.
One of the goals of the models described in [5, 6] is to understand the balance
between pro-inflammatory cytokines such as TNF-α and anti-inflammatory
cytokines such as EPO. We can use the mathematical models together with
computational techniques to help understand this balance by exploring how
changes in parameters related to different aspects of TNF-α and EPO dy-
namics simulate different types of behavior in cartilage injury response. This
paper is devoted to such an exploration. In particular, we would like to
know how changes in parameter values corresponding to different properties
of the TNF-α/EPO interactions influence the lesion expansion or abatement
properties during cartilage injury response.
The remainder of this paper is organized as follows. The next section
provides a brief description of a mathematical model, described fully in [5,
6]1, to which, in this paper, we apply computational methods to explore
some issues regarding the behavior of chondrocytes during the typical injury
response in articular cartilage. It is in that section where we establish ideas
and notation that is used throughout the remainder of this work. The third
section, the results section, shows the computational results and discusses
1We note that there is a slight difference between the models in [5], and in [6]. In this work we use the model in [5] as it gives the same (qualitative) results but replaces a dis- continuous term with a continuous term, and also replaces a phenomenological parameter with one that is more directly connected to the biology.
4
their significance. The paper ends with conclusions drawn from the results
section.
2 Materials and Methods
Here we briefly summarize the mathematical model, established in [5, 6],
used to obtain the computational results of the next section. During in-
jury, chondrocytes are considered as being in specific and distinct “states”
corresponding to which cytokines the cells are capable of producing and re-
sponding to. We refer to the normal state of a subpopulation of chondrocytes
as the healthy state. As a result of inflammation and injury, healthy chon-
drocytes can enter into a “sick” class in which they are at risk of undergoing
programmed cell death. The sick cells are considered as being in one of two
states:
2. the EPOR active state
Cells in the catabolic state are characterized by their ability to produce
TNF-α, while EPOR active cells are characterized by their ability to express
a receptor for EPO. We note that these two cell states are distinct in that
cells capable of producing TNF-α are not capable of expressing the EPO
receptor, and vice versa. Another consequence of cells being in the catabolic
state is that they produce reactive oxygen species (ROS) which serves as a
5
catalyst for the production of EPO by cells in the healthy state.
Due to the fact that there are two typical means of cell death: necrosis,
and programmed cell death known as apoptosis, we also consider two states
for the “dead” class of subpopulations of chondrocytes. We note that for the
purposes considered herein, apoptotic cells do not feed back into the system.
Due to the abrupt nature of the injury, we assume that the initial injury
results in necrosis of cells at the injury site. Furthermore, we assume that cell
death due to secondary cytokine-induced injury is strictly through apoptosis.
The reasoning here is that necrosis is a nonspecific event that occurs in cases
of severe pathological cell and tissue damage, whereas secondary cytokine-
induced injury corresponds with a physiologic form of cell death used to
remove cells in a more orderly and regulated fashion and there is evidence
that often, this is via apoptosis [4].
The typical injury response can be summarized as follows. An injury
results in cell necrosis and the release of alarmins (such as damage-associated
molecular pattern molecules DAMPs), which initiate the chemical cascade
associated with the innate immune and cartilage injury responses [2, 7]. The
DAMPs signal healthy cells near the injury to enter the catabolic state,
catabolic cells are capable of the production of TNF-α which is fundamental
to inflammation. The inflammatory cytokine TNF-α has multifold effects on
the system: It
1. feeds back to promote further switching of cells in the healthy state
into cells in the catabolic state,
6
2. causes cells in the catabolic state to enter the EPOR active state, in
which they express a receptor for EPO and are no longer capable of
synthesizing TNF-α [3],
3. influences apoptosis of cells in the catabolic and EPOR active states,
4. degrades extracellular matrix (denoted by U) which results in increased
concentrations of DAMPs,
5. has a limiting effect on production of EPO [3].
Catabolic cells also produce reactive oxygen species (ROS) which influences
the production of EPO by healthy cells. We denote the concentration of
ROS at a given time and location by R. There is a time delay of 20–24 hours
before a healthy cell signaled by ROS will begin to produce EPO [3].
In the following we use the notation, as in [5, 6], for the mathematical
model of chondrocyte/cytokine interactions during injury response:
1. R - concentration of reactive oxygen species (ROS) at a given time and
spatial location
2. M - concentration of alarmins (DAMPs) at a given time and spatial
location
3. F - concentration of the pro-inflammatory cytokine TNF-α at a given
time and spatial location
7
4. P - concentration of the anti-inflammatory cytokine EPO at a given
time and spatial location
5. U - density of extra-cellular matrix at a given time and spatial location
6. C - population density of healthy cells at a given time and spatial
location
7. ST - population density of catabolic cells at a given time and spatial
location
8. SA - population density of EPO receptor (EPOR) active cells at a given
time and spatial location
9. DN - population density of necrotic cells at a given time and spatial
location
8
The equations making up the mathematical model developed in [5, 6] are
∂tR =∇ · (DR∇R)− δRR + σRST , (1a)
∂tM =∇ · (DM∇M)− δMM + σMDN + δUU F
LF + F , (1b)
∂tP =∇ · (DP∇P )− δPP + σPC(t− τ2) R(t− τ2)
LR +R(t− τ2)
LF + F (t− τ1) − νST
F
LF + F (t− τ1) − αSA
P
LF + F , (1g)
∂tDN =− ηDN , (1h)
∂tU =− δUU F
LF + F . (1i)
Table 1 describes the meaning and units of the model parameters. The
baseline parameter values for the model appear in table 1 of [6]. By base-
line we mean values that are either taken from the literature, or fit to give
quantitative or qualitative agreement with biological observations.
9
δU Rate of Degradation of ECM by TNF-α 1 day
σR Production Rate micromolar·cm2
day·cells
day·cells
day·cells
day·cells
KF Rate limiting concentration for TNF-α micromolar KP Rate limiting concentration for EPO micromolar LR Saturation constant for ROS micromolar LM Saturation constant for DAMPs micromolar LF Saturation constant for TNF-α micromolar LP Saturation constant for EPO micromolar α Response rate of EPOR active cells to EPO 1
day
β1 Response rate of healthy cells to DAMPS/EPO 1 day
β2 Response rate of healthy cells to TNF-α/EPO 1 day
γ Response rate of catabolic cells to TNF-α 1 day
η Rate of degradation of necrotic cells 1 day
ν Response rate of catabolic cells to TNF-α/DAMPs 1 day
µSA Response rate of EPOR active cells to TNF-α 1
day
τ1 time delay in catabolic response days τ2 time delay in production of EPO days
Table 1: Description and units of the parameters appearing in the model (1a)-(1i).
10
In order to compare the simulation results with in vitro observations
it is useful to choose a “measurable”, i.e. a quantity, that can be derived
from results using the model, and easily measured from experiment. Here
we consider the radius of attenuation, this is defined to be the smallest
radius beyond which a lesion cannot expand due to the actions of EPO. In
the computational simulations, it is observed that the radius of attenuation
varies with the change in parameter values. In the following, we will compute
the radius of attenuation as certain specific parameters are varied. To remain
consistent with experiment, we consider injuries to a piece of circular cartilage
of diameter 2.5cm and a time frame of about ten days. In each of the results
discussed below we choose a pair of parameters, then use the mathematical
model to compute how the radius of attenuation varies, as the given pair of
parameters is varied in a systematic way.
How the radius of attenuation varies as dependent on a given pair of
parameters tells us the influence of those parameters on the lesion expansion,
or abatement during cartilage injury response. Based on this information we
gain insight into which aspects of the chondrocyte/cytokine interactions are
most relevant to target in potential therapies. This is one of the principal
motivations for the development of the mathematical model in the first place.
Furthermore, when one parameter in the given pair corresponds to a TNF-α
related term and the other the associated EPO term, we gain insight into
the details of the TNF-α/EPO balancing act discussed in [3].
11
3 Results and Discussion
For all of the following simulations, as in [5, 6], we choose the spatial domain
to be a circle of radius 2.5 cm. This is biologically reasonable since articular
cartilage is divided into three zones [16], with the zone forming the surface of
cartilage, the superficial zone, containing the highest cell density [16]. Fur-
thermore, we assume circular symmetry, since the diffusion of the cytokines
tend to be in the radial direction. This allows for the system (1) to be re-
duced to a problem in one spatial dimension. We choose initial conditions
to represent an initial injury occurring at the center of the domain covering
a disc of radius 0.25 cm. This is typical of the types of impact experiments
that are often performed in orthopaedics labs. The boundary conditions are
taken to be no-flux, i.e.
∂W
∂r
r=2.5
= 0 (2)
for W = R,M, F, P, C, ST , SA, DN , U . This essentially states that the cy-
tokines are confined domain, and are only removed through natural decay
processes. We note that since the system (1) contains delay terms we must
specify not only a condition at time t = 0 but also a history for some time
interval (−T, 0). For time values less than zero, the time of the initial injury,
we take the history to correspond to no injury, i.e. the total cell population
is in the healthy state.
To carry out numerical approximations of the system (1) we discretize in
12
space as follows. Consider the diffusion equation with circular symmetry in
conservative, or divergence, form
where J = D ∂u
∂r is the flux, and D is the diffusion coefficient. Partition the
radii as ri, i = 0, . . . n by dividing the circle into concentric annuli. Then for
0 < i < n we discretize (3) by the formula
π (
1
2
2
. Thus, the scheme (7) corresponds
to the standard finite difference approximation in polar coordinates, see for
example [13].
For the case i = 0, that is, at the center of the circle, we have
πr21 2
∂u(0, t)
2
. (11)
Finally, for a no-flux boundary condition as in (2), the differencing is
given by
These formulas are reproduced from appendix C of [1]2
Applying (7), (11), and (13) to the spatial terms in (1a),(1b),(1c), and
(1d) then gives a semi-discrete system of delay-differential equations
∂tRi =i − δRRi + σR(ST )i, (14a)
∂tMi =i − δMMi + σM(DN)i + δUUi
Fi
LR +Ri(t− τ2)
Fi
LF + Fi(t− τ1) − α(SA)i
, (14i)
for i = 0, . . . , n, where i is the appropriate discrete circularly symmetric
2We note that in [1] there are misprints in the formulas corresponding to (12), (13) which have here been corrected.
15
diffusion operator from (7),(11), or (13). The semi-discrete system is solved
in MATLAB using the delay-differential equation solver dde23. For details
on the methods and software for solving delay-differential equations see [9,
10, 11].
The primary parameter pairs of interest are the time delays τ1, τ2, the dif-
fusion coefficients DF , DP for TNF-α and EPO respectively, the production
rates σF , σP for TNF-α and EPO respectively, and the saturation constants
KF , KP for TNF-α and EPO respectively. These are the parameters that are
most closely tied to the balancing act between pro- and anti-inflammatory
cytokines, and this is what is of primary interest to researchers working to
develop therapies to minimize the collateral damage associated with inflam-
mation in cartilage injuries.
The first pair of parameters we vary are the time delay parameters τ1, τ2.
We recall that τ1 is the delay that for catabolic cells signaled by TNF-α
to become EPOR active, while τ2 is the delay for a healthy cell signaled by
reactive oxygen species (ROS) to synthesize EPO. The baseline values for the
delays are 12 hours for τ1 and 24 hours for τ2 [3, 6]. Figure 1 shows the radius
of attenuation as it varies with τ1, τ2 over the domain [0, 10]×[0, 10] with units
in days. We observe that the delay parameter τ2 has a more significant impact
on the radius on attenuation than does τ1. Since τ2 corresponds to the delay
in a healthy cell signaled by reactive oxygen species to produce EPO, our
results support the hypothesis in [3] that intervention with exogenous EPO
is an important step in limiting the amount of collateral damage caused by
16
TNF-α.
R ad
iu s
Figure 1: Radius of attenuation as it varies simultaneously with τ1, τ2, with other parameters held fixed.
Now we consider the effects of varying the diffusion coefficients DF , DP
for TNF-α and EPO respectively. We note that the diffusion of cytokines
is the principal mechanism that determines the spatial behavior of lesion
spreading in articular cartilage. The baseline values for DF , DP are 0.05,
0.005 mm2
day respectively [6, 8]. Figure 2 shows the radius of attenuation as
a function of DF , DP over the domain [0, 0.1]× [0, 0.015]. We observe that,
of the two diffusion parameters, DF has the greater impact on the radius of
attenuation. This is somewhat expected in light of the fact that there is a
time delay for production of EPO by healthy cells signaled by ROS. Because
of this delay TNF-α is typically produced at significantly earlier times than
17
EPO. Thus the degree to which TNF-α can diffuse significantly influences
how far the lesion can spread during this initial time period before there are
sufficient concentrations of EPO to abate the spread of damage.
0
0.05
0.1
0
0.005
0.01
s
Figure 2: Radius of attenuation as it varies simultaneously with DF , DP , with other parameters held fixed.
Next, we consider the radius of attenuation as it varies simultaneously
with the saturation constants KF and KP . The baseline values for these
parameters are 10 and 1 respectively, with units in micromoles. Figure 3
shows the radius of attenuation as a function of KF , KP over the domain
[0, 100] × [0, 100] with units in micromoles. This is quite a large variation
for KF and KP . We observe that the saturation constant, KP , for EPO to
limit the response of healthy cells to TNF-α and alarmins (DAMPs) has the
greater influence of the two parameters KF , KP on the radius of attenuation.
18
Thus the results of the mathematical model seem to suggest that healthy
cells must be sensitive to relatively low concentrations of EPO in order to
minimize damage, and for maximal healing to occur. Figure 4 again shows
the radius of attenuation as a function of KF , KP but for a smaller range in
the parameter values. Here we focus on values relatively close to the baseline
values, this gives an idea of how sensitive the model is to small, simultaneous
changes in values for the parameters KF , KP . The results are consistent with
those shown in figure 3.
0
50
100
0
50
s
Figure 3: Radius of attenuation as it varies simultaneously with KF , KP , with other parameters held fixed.
We observed that when comparing the saturation parametersKF , KP , the
parameter KP has the more significant effect on the radius of attenuation.
However, the term involving KP and TNF-α in the model system (1a)-(1i)
19
s
Figure 4: Radius of attenuation as it varies simultaneously with KF , KP , with other parameters held fixed.
is
KP + P , (15)
which influences the switch from the healthy to the catabolic state. Thus, the
switching is determined by the parameters β2 and KP together. We examine
the radius of attenuation as a function of β2, KP , the results are shown in
figure 5. Again it is observed that KP has the more significant impact in
determining the radius of attenuation.
We next examine the radius of attenuation as as it varies simultaneously
with the production rates σF , σP of TNf-α and EPO respectively. The base-
line values for these parameters are 0.0001 for σF and 0.001 for σP with units
in days−1. Figure 6 shows the radius of attenuation as a function of σF , σP
20
40
50
60
5
10
15
R ad
iu s
Figure 5: Radius of attenuation as it varies simultaneously with β2, KP , with other parameters held.
over the domain [0, 0.0001]× [0, 0.1]. We observe that, overall, the radius of
attenuation increases as σF , the production of TNF-α, increases. However,
the radius of attenuation appears to be a nonlinear function of σF , σP . In
figure 7 we show the radius of attenuation as a function of σF , σP over the
domain [0, 0.0003]× [0.001, 0.005]. This shows that, while the radius of at-
tenuation is generally increasing as a function of σF , that for a fixed value
of σP there is a point beyond which the radius of attenuation exceeds the
domain. Thus, for our domain, if σF is sufficiently large then there is no ra-
dius of attenuation. However, this does not necessarily imply that no healing
whatsoever can take place.
Figure 8 shows the dynamics of the healthy and penumbral (sum of
21
sigmaF sigmaP
R ad
iu s
Figure 6: Radius of attenuation as it varies simultaneously with σF , σP , with other parameters held.
1 2
s
Figure 7: Radius of attenuation as it varies simultaneously with σF , σP , with other parameters held fixed.
22
catabolic and EPOR active) cell populations for values of σF and σP that,
according to figure 7, lead to a radius of attenuation of approximately 1.7cm.
Since the radius of attenuation in figure 7 is computed based on a ten day
time period, figure 8(i) shows the cell populations as a function of radius after
a twenty day period to ensure that the radius does not continue to expand
after ten days. We observe that, while the radius of attenuation is larger than
that in figure 6 of [6], there is still at least as much healing near the initial
injury site due to EPOR active cells switching back to healthy as a result of
EPO signaling. An interesting observation is the “dip” in the healthy cell
population between radius r = 0.25cm and r = 0.7 if figure 8(d). This is due
to the fact that diffusion of TNf-α results in lower concentrations of TNF-
α for smaller radius values where there is a higher concentration of EPOR
active cells. Thus, the EPOR active cells can more effectively response to
EPO.
Figure 9 shows the dynamics of the healthy and penumbral (sum of
catabolic and EPOR active) cell populations for values of σF and σP that,
according to figure 7, lead to no radius of attenuation, that is, there is no
point in our domain for which secondary TNF-α induced damage cannot
spread. Again, we point out that this does not imply that no healing occurs.
We see in figure 9(i) that after twenty days there is a significant healthy cell
population despite that the penumbra spread throughout the entire domain.
We again see in figure 9(i) the “dip” in the healthy population which is now
more pronounced than in figure 8(d).
23
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
Figure 8: Cell populations when σF = 0.00017 and σP = 0.0032.
24
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
Figure 9: Cell populations when σF = 0.0004 and σP = 0.003.
25
We make one further observation. If the production rate σF of TNF-α
is set to zero, according to the equations in system (1a)-(1i) there should
be a penumbra made up entirely of catabolic cells. Figure 10 shows the cell
population density profile for healthy and catabolic cells after ten days. We
see that it is indeed the case that there is a penumbra made up entirely of
catabolic cells. This result suggests that a delay in the production of TNF-
α could allow for the build up of a large population of catabolic cells, so
that, once TNF-α is produced there will be a wave of apoptosis and EPOR
activation. Depending on the concentration of EPO available this scenario
could lead to more damage than is typical. Thus, damage associated with a
delay in the production of TNF-α could potentially be worse in some cases
than is seen with just the delay in the production of EPO.
0 0.5 1 1.5 2 2.5 0
2
4
6
8
Figure 10: Cell populations after 10 days with σF = 0.
26
4 Conclusions
The work in the previous section is an incomplete exploration of parameter
space. We note however that it does suggest that the ratio σP
σF between the
EPO production rate and the TNF-α production rate plays a significant role
in determining the radius of attenuation. The results shown in figures 8, and
9 imply that EPO, or more generally the anti-inflammatory arm of cartilage
injury response is robust. Even in cases where the ratio σP
σF is small but
nonzero inflammation does not result in the uncontrolled spread of injury as
in the case when σP = 0. It is ultimately desirable to derive theoretical results
that give complete detailed knowledge of the qualitative behavior of solutions
to system (1a)-(1i) as a function of the parameter values. This is a difficult
problem due to the number of equations and large number of parameter val-
ues. One future direction for the work presented above is its application to
real experiments. It is common in orthopaedics research to perform impact
experiments on large animal joints, typically bovine or porcine, or harvested
human joints in attempts to replicate cell-level pathology in intra-articular
fractures, see e.g. [15, 14]. It is likely that cytokine measurements relevant
to the work presented here can be made from such experimental studies. An-
other future direction for the work presented here and in [5, 6] is to include
mechanical effects that are important in cartilage injury. Of particular inter-
est is the representation of effects associated with shear stress to cartilage. In
general, articular cartilage in joints such as the knees and ankles can respond
27
efficiently to direct impact mechanical stress. However, cartilage is less re-
sistant to shear stress. A mathematical and computational models that are
capable of giving insight into what happens when shear stress is applied will
be of great value to orthopaedics research.
5 Acknowledgements
The author would like to thank Bruce Ayati, Jim Martin, Prem Ramakrish-
nan, and Lei Ding for valuable discussions regarding this work. The author
would like to express sincere gratitude to the editor and reviewers for their
valuable comments and suggestions.
[1] B.P. Ayati. Methods for computational population dynamics. ProQuest
LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–The University of Chicago.
[2] M. E. Bianchi. Damps, pamps and alarmins: all we need to know about
danger. J. Leukoc. Biol., 81:1–5, 2006.
[3] M. Brines and A. Cerami. Erythropoietin-mediated tissue protection:
reducing collateral damage from the primary injury response. J. Intern.
Med., 264:405–432, 2008.
28
[4] M. Del Carlo and R.F. Loeser. Cell death in osteoarthritis. Current
rheumatology reports, 10(1):37–42, 2008.
[5] J.M. Graham. Mathematical representations in musculoskeletal physi-
ology and cell motility. ProQuest LLC, Ann Arbor, MI, 2012. Thesis
(Ph.D.)–The University of Iowa.
[6] J.M. Graham, B.P. Ayati, L. Ding, P.S. Ramakrishnan, and J.A. Mar-
tin. Reaction-diffusion-delay model for epo/tnf-α interaction in articular
cartilage lesion abatement. Biology Direct, 7(9), 2012.
[7] H. E. Harris and A. Raucci. Alarming news about danger. EMBO
Reports, 7(8):774–778, 2006.
[8] H.A. Leddy and F. Guilak. Site-specific molecular diffusion in articu-
lar cartilage measured using fluorescence recovery after photobleaching.
Annals of Biomedical Engineering, 31(7):753–760, 2003.
[9] L. F. Shampine and M. W. Reichelt. The matlab ode suite. SIAM J.
Sci. Comput., 18(1):1–22, 1997.
[10] L. F. Shampine and S. Thompson. Solving ddes in matlab. Appl. Numer.
Math., 37(4):441–458, 2001.
[11] L. F. Shampine and S. Thompson. Numerical solution of delay differ-
ential equations. Springer, New York, 2009.
29
[12] J.A. Sherratt and J.D. Murray. Models of epidermanl wound healing.
Proc. Bio. Sci., 241(1300):29–36, 1990.
[13] J. W. Thomas. Numerical partial differential equations: finite difference
methods, volume 22 of Texts in Applied Mathematics. Springer-Verlag,
New York, 1995.
[14] Y. Tochigi, P. Zhang, M.J. Rudert, T.E. Baer, J.A. Martin, S.L. Hillis,
and T.D. Brown. A novel impaction technique to create experimental
articular fractures in large animal joints. Osteoarthritis and Cartilage,
21(1):200 – 208, 2013.
[15] Yuki Tochigi, Joseph A. Buckwalter, James A. Martin, Stephen L. Hillis,
Peng Zhang, Tanawat Vaseenon, Abigail D. Lehman, and Thomas D.
Brown. Distribution and progression of chondrocyte damage in a whole-
organ model of human ankle intra-articular fracture. The Journal of
Bone & Joint Surgery, 93(6):533–539, 2011.
[16] M. Ulrich-Vinther, M.D. Maloney, Schwarz E.M., Rosier R., and R.J.
O’Keefe. Articular cartilage biology. J. Am. Acad. Orthop. Surg.,
11(6):421–430, 2003.